The volume of a cylinder is given by the formula v = π(r^2)h. find the maximum value of v if r + h = 12

V = 256π unit^3.

Step-by-step explanation:

V = π(r^2)h

r + h = 12

So h = 12 - r.

Substituting for h in the formula for V:

V = π r^2(12 - r)

V = 12πr^2 - πr^3

To find the  value of r when V is a maximum we find the derivative with respect to r:

V' =  24πr - 3πr^2

This equals 0 for maximum/minimum:

24πr - 3πr^2 = 0

3πr(8 - r) = 0

r = 0 or 8.  (We ignore the 0).

The second derivative is  24π - 6πr which is negative when r = 8 so r =8 gives a maximum value for V.

Therefore the maximum value of V is π(8)^2h

h = 12 - 8 = 4 so

Maximum V = 64*4π

V = 256π unit^3.

a reflection across the x-axis , followed by the by a reflection across the y-axis , and then a scale dilation of 0.5 .